nonsymplectomorphic

Nonsymplectomorphic is an adjective used in symplectic geometry to describe a pair of symplectic manifolds that are not related by a symplectomorphism. A symplectomorphism is a diffeomorphism between smooth manifolds that pulls back one symplectic form to the other, preserving the nondegenerate 2-form structure. Thus, two manifolds with symplectic forms ω1 and ω2 are nonsymplectomorphic if there is no diffeomorphism f such that f*ω2 = ω1.

The term is typically applied in discussions of classification and comparison of symplectic structures. It highlights that the symplectic category is finer than the smooth category: even when two manifolds are smoothly the same, their symplectic structures may be inequivalent under symplectomorphisms.

Examples and usage patterns: On some 4-manifolds, there exist distinct symplectic forms that are not related by a diffeomorphism preserving the forms, making the corresponding symplectic manifolds nonsymplectomorphic. This frequently occurs on blow-ups of complex projective spaces or other complicated smooth manifolds, where different choices of symplectic forms lead to non-symplectomorphic structures despite identical underlying topology. It is also common to show nonsymplectomorphism by invoking obstructions from invariants such as Seiberg-Witten invariants, Gromov–Witten theory, or symplectic capacities, which can distinguish symplectic forms beyond cohomology.

Related concepts: symplectic form, symplectomorphism, Seiberg-Witten invariants, Gromov–Witten invariants, symplectic capacity. The term is a shorthand used to assert nonexistence of a symplectomorphism between two given symplectic manifolds.